We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$ its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in $\ell$ and $n$. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any. Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

中文翻译：

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计算没有高度的空缺多项式的多重线性因子

我们提出了一种确定性算法，该算法计算数域上多元缺失多项式的多重线性因子。它的复杂性是 $\ell^n$ 中的多项式，其中 $\ell$ 是输入多项式的空白大小，$n$ 是变量的数量，特别是其次数的对数多项式。我们还为 $\ell$ 和 $n$ 中复杂度多项式的相同问题提供了随机算法。在其他特征零域和大特征有限域上，我们的算法计算具有至少三个多元多项式单项式的多重线性因子。提供了下限来解释我们算法的局限性。作为副产品，我们还为未知的多项式族设计了多项式时间确定性多项式恒等式检验。我们的结果基于所谓的间隙定理，它将高度分解减少到重复的低度分解。虽然以前的算法使用以系数的高度表示的间隙定理，但我们的间隙定理仅取决于多项式的指数。这使得我们的算法更加基本和通用，并且在大多数情况下更快。